In the zweibein formulation, the Carroll-Weyl transformation defined in [1]} for general \(z\) , can be written as
\(e^0_{\mu } \rightarrow e^{z\Omega (\tau ,\sigma )}e^0_{\mu }, \quad h_{\mu \nu } \rightarrow e^{2\Omega (\tau ,\sigma )}h_{\mu \nu },\) 
